This section provides the notational conventions used throughout this thesis.

A *digital image*
on a 2D
finite arithmetic lattice
is a function
that maps the supporting
lattice
onto a finite set
of signal values (e.g., grey levels, colours, or multi-band
signatures). For simplicity, the set
is assumed to be
integer-valued in the range between 0 and .

A pixel in an image is denoted by where is the positioning index in the lattice . The set of all pixels in an image with exception of a pixel is denoted by . Similarly, all sites on lattice except the th site is denoted by , so that .

The notation
defines a *neighbourhood* of a
site , which usually consists of a set of sites with a specific
spatial configuration. All sites in a neighbourhood are dependent of
each other.

Because the thesis deals mainly with probability texture models, in particular Markov-Gibbs random fields, related terminology and notations are presented below.

A random variable has a nondeterministic value with a given
probability distribution. A sequence or array of statistically
interrelated random variables form a discrete stochastic process.
Let a random variable
be associated
with each site
. Then these variables form a
*random field*
in a
*configuration space*
if a probability measure
exists. A
random field is a two-dimensional stochastic process. The
configuration space has a combinatorial number of different images,
.

Under a random field model, an image is considered as an
instance or a sample of the configuration space
. The
image probability , or simply, , is known as the
*joint probability* of the image signals. Another commonly-used
probability measure is *conditional probability* of a pixel
given all other pixels , denoted by
. The conditional probability is a local probability of image
signals, while the joint probability is a global one.

For convenience's sake, the notational conventions and symbols used in this thesis are summarised in Table 4.1.

Notation | Meaning |

A 2D finite arithmetic lattice | |

The set of integral signal values | |

A digital image defined on | |

All pixels in the image except the pixel | |

A random field | |

The configuration space of a random field | |

Neighbourhood of the site | |

The lattice with exception of the th site | |

All sites of with exception of a site | |

Joint probability of a sample image | |

Conditional probability of image signal at pixel given |